Testing sequence for boundaries and monotony

[Juan Pablo]

I'm trying to solve a problem from a practice test I was given. It says. Determine if the sequence is bounded and if it's decreasing or increasing:
$$a = \frac{arcsin(1/n)}{arctan(1/n)}$$

I don't really know where to start as I don't remember seeing how to test sequences for boundaries or anything like that and there's nothing on my book saying how to test it in the sequences chapter.

If anyone is interested, the practice test: http://profesores.usfq.edu.ec/valen/departamentomatematicas/files/exapractica/prac_destr_mat132/index.htm

Spanish, sorry

Any hint?

Thanks!

 

[Dick]

The problem is really asking you about the behavior of the function f(x)=arcsin(1/x)/arctan(1/x) as x->infinity. The bounded part involves showing whether lim f(x) as x->infinity has a limit. Do you know l'Hopital's rule? For the monotone part can you show f'(x) has a constant sign for x>1? I think that's the hard part. I don't think it's a terribly easy problem.

 

[lanedance]

Hi Juan Pablo

bounded from above means there exists some N such that a_n < N for all n

the sequence is monotonically increasing if a_n-a_n-1>0 for all n

hope this helps you get started

 

[Juan Pablo]

Thanks to you both. I helped a lot!